Collective behavior of soft self-propelled disks with rotational inertia

We investigate collective properties of a large system of soft self-propelled inertial disks with active Langevin dynamics simulation in two dimensions. Rotational inertia of the disks is found to favor motility induced phase separation (MIPS), due to increased effective persistence of the disks. The MIPS phase diagram in the parameter space of rotational inertia and disk softness is reported over a range of values of translation inertia and self-propulsion strength of the disks. Our analytical prediction of the phase boundary between the homogeneous (no-MIPS) and MIPS state in the limit of small and large rotational inertia is found to agree with the numerical data over a large range of translational inertia. Shape of the high density MIPS phase is found to change from circular to rectangular one as the system moves away from the phase boundary. Structural and dynamical properties of the system, measured by several physical quantities, are found to be invariant in the central region of the high density MIPS phase, whereas they are found to vary gradually near the peripheral region of the high density phase. Importantly, the width of the peripheral region near the phase boundary is much larger compared to the narrow peripheral region far away from the phase boundary. Rich dynamics of the disks inside the high density MIPS phase is addressed. Spatial correlation of velocity of the disks is found to increase with rotational inertia and disk hardness. However, temporal correlation of the disks’ velocity is found to be a function of rotational inertia, while it is independent of disk softness.


Model and methods
We perform Langevin dynamics 30 simulation on N = 48,400 self-propelled disks in a commensurate rectangular box of dimensional ratio L x /L y = 2/ √ 3 with periodic boundary conditions using GPU and CPU versions of the Molecular Dynamics solver MPMD 21,27,39,40 . Each disk with diameter σ , mass m, and moment of inertia I interacts with other disks in the system through a modified Yukawa potential 21,27 , such that the total conservative potential is U = U 0 i<j e −(r ij −σ )/ /r ij . U 0 is the interaction strength and controls the softness of the disks.
Dynamics of the center of mass r i and the self-propulsion direction θ i of the disks are governed by the set of N active Langevin equations: We have neglected thermal noise of the translational motion in this study, as the thermal noise is orders of magnitude smaller 22,34,41 compared to the other forces in Eq. (1). The suffix r denotes rotational parameter. γ , γ r are friction coefficients. D r is the angular diffusion coefficient of the direction of self-propulsion n i = (cos θ i , sin θ i ) . ξ is the Gaussian white noise with zero mean and unity variance. F I i is total conservative force on disk i, such that F I i = −∇ i U . The self-propulsion force F a i is responsible for the motility of the disks. The disks move persistently with constant speed v 0 along the self-propulsion axis n i for an average time τ p = 1/D r . The average time τ p is the persistence time of the disks. We consider σ and 1/D r as the normalized unit for length and time, respectively. Energy is measured in the unit of mσ 2 D 2 r . The normalized dynamical equations are: M = m/γ 1/D r and J = I/γ r 1/D r are the reduced translational and rotational inertia, respectively. Strength of activity of the disks is controlled by the Peclet number P e = v 0 /D r σ . In normalized units, Peclet number is the ratio of persistence length v 0 /D r to the diameter σ of the self-propelled disks. Hence, the Peclet number scales with the activity of the disks. Ŵ = U 0 γ σ 3 D r and κ = σ are the reduced interaction strength and softness parameter of the disks, respectively. With decrease in κ , softness of the disks increases 21 , whereas hardness increases with increase in κ.
Area fraction φ = Nπ/4L x L y is fixed at 0.65, as MIPS is observed without rotational inertia for the considered value of φ . We consider constant interaction strength Ŵ = 25 , which is found to be an optimum value for the complete range of softness parameter κ in this work, as the value above and below Ŵ = 25 shifts the MIPS phase boundary towards higher κ , similar to that in Ref. 21 21 . Interaction cut-off is set to r c = 8.0 , which is significantly large compared to the range of inter-particle interaction of the particles. Integration time step δt is set to 10 −4 .

Results and discussion
Several interesting collective properties of the self-propelled inertial disks are provided and discussed below.
MIPS phase diagram. The disks are distributed almost homogeneously in the limit of large softness ( κ → 2 ). Nearly homogeneous distribution of the disks is called as homogeneous phase. At sufficiently large value of softness parameter κ and rotational inertia J, the self-propelled system is found to phase separate into a high density and a low density region, that coexist. The coexistence of high and low density region is known as MIPS. In Fig. 1, the phase diagram of our system of soft self-propelled disks in the space of rotational inertia J and softness parameter κ in semi-log scale for fixed translational inertia M = 0.05 and Peclet number P e = 125 is shown.
Green (triangles) and red (circles) markers denote, respectively, the MIPS and the homogeneous phase. MIPS state in the phase diagram is obtained from two distinct peaks in the distribution of local area fraction. In the limit of small values of J ( J < 0.06 ), the homogeneous system of soft disks is found to exhibit MIPS at sufficiently large κ ( κ > 9 ). In the small J limit, the phase boundary between the homogeneous and the MIPS state is independent of J. In the limit of large values of J ( J ≥ 0.06 ), the collection of self-propelled disks are found to exhibit MIPS at significantly small values of κ or large softness. In the limit of large J, phase boundary (3) www.nature.com/scientificreports/ monotonically shifts towards small κ with increase in J. Unlike softness and translational inertia, which in general oppose MIPS, rotational inertia is found to favor MIPS such that MIPS is observed for extremely soft disks with increase in rotational inertia.
To understand the phase diagram, more specifically the shape of the phase boundary, a time scale τ κ is associated to the deformation of the soft disks. The softness timescale τ κ is defined as τ κ = τ P /κ . The hard disks at large κ are less deformed despite persistent collision with the other disks. Consequently, τ κ is small for hard disks, whereas the soft disks at small κ deform significantly, that in turn increases τ κ . Moreover, τ κ increases with Peclet number P e , as the soft disks are deformed more with increase in P e . As the deformation of the disks increases with increase in τ κ , the effective size of the two-particle-cluster becomes smaller. Interestingly, with increase in J, temporal correlation of the self-propulsion direction n i increases, as shown in the inset of Fig. 7a. Hence, the effective persistence time τ e P of the self-propelled disks increases with increase in rotational inertia J. In the limit of small and large J , that is in the limit of small rotational inertial time scale τ I = I/γ r with respect to persistence time scale τ P and vice-versa,, the analytical expression of τ e P as obtained by Caprini et al. 37

is
The system of soft self-propelled disks phase separate when the effective persistence time becomes sufficiently large such that the small two-particle-cluster remain together for sufficiently large time for the other disks to collide with the nucleation site of this two-particle-cluster. Hence, the necessary criteria to exhibit MIPS is that the softness time τ κ of soft self-propelled disks should scale with the effective persistence time τ e P , that is τ e P ∼ τ κ . Hence, the scaling of the phase boundary in the appropriate limit of J is obtained as In Fig. 1, the analytical expression for the phase boundary in the limit of small and large J, respectively, are shown by the dashed and the solid line. Our analytic prediction Eq. (6) fits well with the simulation data.
As translational inertia M suppresses and Peclet number P e favors MIPS, the phase boundary between the homogeneous and MIPS states in the J − κ space is found to shift towards larger κ with increase in M and decrease in P e . In Supplementary Fig. S1, the phase diagram in the J − κ space for a smaller value of Peclet number P e = 75 and for three different values of M, spanning over two orders of magnitude, namely M = 0.005 , M = 0.05 , and M = 0.5 is shown. In the small J limit, the horizontal phase boundary shifts towards large κ , that is towards substantially hard disks, with increase in M and decrease in P e . With increase in κ , bounce back effect 20 on collision of two inertial self-propelled disks becomes important. Interestingly, our analytic prediction for the phase boundary matches well with the numerical data, even with variation in M and P e . Due to shift in the phase boundary towards large κ with increase in M and decrease in P e , the proportional constant factor in the analytic expression for the phase boundary should be a function of M/P e . Due to similar variation of the phase boundary in the limit of large J, the proportionality constant should be a function of M/P e . However, the exact dependency of phase boundary on the M/P e is out of the scope of this study.
Shape of the high density phase. In Fig. 2, we have shown the configuration of the system for κ = 6, 8, 10, 11, 18 from left to right for fixed J = 2 , M = 0.05 , and P e = 125 , corresponding to the vertical column at J = 2 in MIPS region of Fig. 1.
If the local area fraction of the particles is greater than the considered global area fraction φ = 0.65 , then those particles are considered to constitute the dense cluster phase, else they are considered to constitute dilute or low density phase. Particles in the high and the low density region is colored magenta and green, respectively. The configurations are obtained at t = 100 . Close to the phase boundary, the shape of the high density phase is found to be nearly circular, while the shape is near rectangular as the system is much away from the phase   www.nature.com/scientificreports/ boundary. Thus, we observe the shape change from circular to a rectangular one as we move away from the phase boundary with increase in κ . It is interesting to note that the shapes of the clusters may vary 13 with variation in the geometry of the simulation box. For simplicity and without loss of generality, in this work, we consider a rectangular simulation box of fixed dimensions. Near the phase boundary, it has been previously 21 observed that the phase separation time increases. To justify the fact, that at large time the shape remains circular near the phase boundary, we show the configuration for J = 2 , κ = 7 , M = 0.05 , and P e = 125 at t = 50 (short), t = 100 (intermediate), and t = 500 (large) in Supplemental Fig. S2. At all three times the high density region is found to be near circular. The circular shape near the phase boundary is found at small as well as large inertia. We show the configuration for the parameters M = 0.5 (large inertia), J = 2 , κ = 18 , and P e = 125 , for which the system is near the phase boundary and in the MIPS state, in supplemental Fig. S4b. The high density phase is near circular despite the hard particle limit at κ = 18 . The detailed analysis of the observed clusters is performed in the following Sections.
Structural properties. Near and away from the phase boundary the local structural properties of the system exhibit very distinctive features. In Fig. 3, we have shown the local structural properties for κ = 7 (top row) and κ = 17 (bottom row) for fixed M = 0.05 , J = 2 , and P e = 125. In Fig. 3a,d, local area fraction 26 φ (left vertical axis) and the magnitude of the local orientational order 21,25 q 6 =< | 1 6 6 k=1 e j6θ(r ik ) | > (right vertical axis) from the center of the high density phase towards the periphery is shown. θ(r ik ) is the angle between the two disks at r i and r k with respect to the x-axis. j is the imaginary number, and < ... > denotes the local average. For κ = 7 , which is near the phase boundary, and for κ = 17 , which is away from the phase boundary, the high density phase is circular and rectangular, respectively. Moreover, for  Configuration of the system. The disks in the high and the low density phase are colored magenta and green, respectively. Boundary of the central and peripheral region for the circular high density phase is denoted by r 1 = 42.5 and r 2 = 80 , respectively, and that for the rectangular high density region is denoted by L 1 = 69 and L 2 = 77 , respectively (see main text for definition of central and peripheral region). (c,f) Configuration of the system, colored according to the magnitude of orientational order q 6 (r i ) of the disks. www.nature.com/scientificreports/ calculational ease the center of the high density phase is shifted to the center of the simulation box. The configuration of the system is shown in Fig. 3b,e for κ = 7 and κ = 17 . The particles in the high and the low density phase are colored magenta and green, respectively. The central and the peripheral region is denoted by the area inside the solid circle (denoted by r 1 ) and the annular area between the solid and the dashed circles (denoted by r 2 ) in Fig. 3b, respectively. For the rectangular cluster, we consider the central and the peripheral region respectively by an inner and an outer rectangular box of width 50 units. Outer boundary of the central (denoted by L 1 ) and the peripheral (denoted by L 2 ) rectangular box is shown by the solid and dashed lines, respectively. Hence, for both the circular and the rectangular clusters, solid and dashed line denote the boundary of the central and the peripheral region, respectively, as in Fig. 3a,d. The boundary of the central and the peripheral region is r 1 ≈ 42.5 , r 2 ≈ 80 for κ = 7 and L 1 ≈ 69 , L 2 ≈ 77 for κ = 17 . In Fig. 3c,f, the magnitude of the orientational order q 6 (r i ) of the disks for κ = 7 and κ = 17 are shown. Local structural properties remain nearly constant in the central region, while they vary gradually in the peripheral region. The width of peripheral region r p = r 2 − r 1 = 37.5 for κ = 7 is much larger compared to that ( r p = 8 ) for κ = 17 . The magnitude of the local area fraction in the central region is sufficiently smaller for the hard disks ( κ = 17 ) compared to that for the soft disks ( κ = 7 ). This is due to the fact that the soft disks deform substantially, whereas the hard disks practically do not deform upon collision with the other disks. Consequently, the central region is jammed for the hard disks, while the soft disks have the liberty to infiltrate towards the center of the central region due to their large softness. Inside the central region, the magnitude of local orientational order is q 6 ≈ 1 for the hard disks, while it is slightly smaller than 1 for the soft disks because of their softness. Angular average of pair correlation function or the radial distribution function RDF(r) 21,27 in the high and the low density phase measures the translational order of the phases. In Fig. 4a, we show the RDF(r) in the high (blue) and the low (red) density phases for the parameters M = 0.05 , J = 2 , and P e = 125 for the soft disks at κ = 6.
The low density phase is clearly fluid-like as there is only a single peak in the RDF(r). There are few peaks in the high density phase for the soft disks at κ = 6 . In the inset of Fig. 4a, we show the RDF(r) for the hard disks at κ = 18 in the high density phase. Not only several RDF(r) peaks are present for κ = 18 , the first few peaks are distinctly divided into two sub-peaks, which is a characteristic of hexagonal crystal 42 in passive two dimensional system. Moreover, the magnitude of the local orientational order of the system is nearly 1 in the central region for both the soft and the hard disks (see Fig. 3a,d). Hence, the system of soft disks resembles a two dimensional hexatic phase and the hard disks resembles a two dimensional crystalline phase 19,27,43 . The radial location r FP of the first peak of RDF(r) in the high density phase is a measure of inter-particle separation 21 of the high density disks. In Fig. 4b, we show the variation of r FP with J (left vertical axis) (circles) for fixed κ = 7 and with κ (right vertical axis) (triangles) for fixed J = 2 . Inertial parameter M and Peclet number P e are fixed at 0.05 and 125, respectively. With increase in disk hardness (increase in κ ), the inter-particle separation increases, which is consistent with the fact that deformation decreases with increase in disk hardness. Importantly, with increase in J, the inter-particle separation decreases by a small amount.
At small inertia M = 0.005 , similar structural properties with wide peripheral region near the phase boundary at small κ and a narrow peripheral region away from the phase boundary in the limit of large κ , analogous to that for M = 0.05 , is observed. At M = 0.005 , the structural properties and the corresponding configuration for κ = 6 (near the phase phase boundary) and κ = 18 (away from the phase boundary) are shown in Supplementary Fig. S3a,b,d,e, respectively, for fixed J = 1 , P e = 125 . However, the width of the peripheral region near the phase boundary at large inertia M = 0.5 is found to be significantly small compared to that at small inertia ( M ≤ 0.05 ). Importantly, with increase in J, the system moves away from the phase boundary, analogous to that with increase in κ . In Supplementary Fig. S4a,b,d,  www.nature.com/scientificreports/ φ ≈ 1.25 for M = 0.05 (Fig. 3a) in the central region. Hence, the less wide peripheral region at large inertia is due to the fact that the phase boundary, in fact, is observed at large κ.
Dynamical properties. We show the local speed 26 v = �(v i · v i ) 1/2 � as a function of distance from the center of the high density phase for κ = 7 in Fig. 5a and κ = 17 in Fig. 5d. Identical values for the other parameters are considered as that in Fig. 3. The boundaries of the central and the peripheral regions are same as shown in Fig. 3. Local speed remains nearly constant in the central region and it increases in the peripheral region. The magnitude of the local speed in the central region is more for the soft disks ( κ = 7 ) than for the hard disks ( κ = 17 ). At an arbitrary initial time t = t 1 , the configuration for κ = 7 and κ = 17 are shown in Fig. 5b,e, respectively. At t = t 1 , the disks are colored green in the central region and yellow in the peripheral region, and the other disks are colored purple. Configuration at a later time t = t 1 + 30 is shown in Fig. 5c,f. From the Fig. 5b,e, it is clear that the dilute phase for κ = 7 is more crowded compared to the dilute phase for κ = 17 . Consequently, the values of the local area fraction is large in the dilute phase for κ = 7 (Fig. 3a) as that for κ = 17 (Fig. 3d). As a result of comparatively large area fraction or more crowding, collisions between the disks are more in the dilute phase for κ = 7 than that for κ = 17 . Hence, the average magnitude of local speed for the disks in the low density phase is less for κ = 7 due to more collisions as that for κ = 17.
Number of disks in a specific region of the high density phase at t = t 1 is denoted by N(t = t 1 ) . Due to the dynamical movement of the disks, let N(t − t 1 ) be the number of disks retained in that particular region, out of the initial N(t = t 1 ) disks, at t − t 1 . To quantify the disk movement in the high density phase, we define the retention fraction 26,27 R(t − t 1 ) = N(t − t 1 )/N(t = t 1 ) for t > t 1 , the number of disks retained at t − t 1 with respect to the number of disks at t = t 1 . In Fig. 6, we show the retention fraction R(t − t 1 ) for κ = 7 (Fig. 6a) and κ = 17 (Fig. 6b).  www.nature.com/scientificreports/ Green and yellow curves correspond to the particles in the central and the peripheral region, respectively, as shown in Fig. 5b,e. In the central region, the local speed becomes small for the soft disks, as the local area fraction becomes large. As the local speed for the soft disks at κ = 7 in the peripheral region is sufficiently large, the particles in the peripheral region mix throughout the system. Hence, the retention fraction of the soft disks in the peripheral region (yellow curve in Fig. 6a) saturate quickly. However, due to their softness, the peripheral disks (yellow disks) and the disks in the low density phase (purple disks) at t = t 1 steadily move towards the center of the high density region. Consequently, the soft disks of the central region at t = t 1 (green disks) steadily shuttle out of it, to maintain the constant area fraction in the central region. Hence, a finite slope is found for the green curve of R(t − t 1 ) at t = t 1 + 30 for the soft disks ( κ = 7 ). At much larger time t − t 1 > 30 , it saturates. Time evolution for the soft disks ( κ = 7 ) is shown in Supplementary Movie 1. On the other hand, due to extremely small deformation and negligible speed of the hard disks ( κ = 17 ), it is practically impossible for the peripheral disks (yellow) and the disks in the low density phase to penetrate deep inside the central region. Small number of disks in the narrow peripheral region shuttle out of the high density phase and facilitate dynamic variation of the boundary of the high density phase. Hence, R(t − t 1 ) corresponding to the peripheral disks at κ = 17 saturates quickly at small value ( R(t − t 1 ) → 0 ). Peripheral disks (yellow colored disks) rejoin the high density phase through the other side, due to periodic boundaries. A finite fraction of disks still penetrate inside the central region only due to dynamic variation of the boundary or the overall dynamical variation in shape of the high density region. Hence, R(t − t 1 ) for the hard disks in the central region nearly saturates at t − t 1 ≈ 27 . Dynamical variation of the system of hard disks κ = 17 is shown in Supplementary Movie 2. In the phase separated state at small ( M = 0.005 ) and large ( M = 0.5 ) inertia similar dynamical properties, as for M = 0.05 , is observed (see Supplementary Figs. S3c,f and S4c,f. With increase in rotational inertia J, the local speed of the soft disks in the central regions increases, due to increase in effective persistence time τ e P of the disks. In spite of the increase in local packing fraction of the soft disks with increase in J, the peripheral disks and the disks in the low density phase at an arbitrary time t = t 1 move to the center of the high density phase more quickly, due to increase in local speed in the central region with J. In Supplementary Fig. S5, we compare the local speed (left vertical axis) and the local area fraction (right vertical axis) in the MIPS state for two different values of J, namely J = 2 and J = 10 , for fixed M = 0.05 , κ = 7 , and P e = 125 . Local speed in the central region is much larger for J = 10 compared to that for J = 2 . In Supplementary Movie 3 we show the dynamic evolution of the system at large rotational inertia ( J = 10 ) and for fixed parameters M = 0.05 , κ = 7 , and P e = 125.
Temporal and spatial correlations. In the low density phase, the velocity of the self-propelled disks v i align with the self-propulsion direction n i of the disks, due to infrequent interaction with the other disks. However, in the high density phase, due to frequent collisions with the neighboring disks, alignment between v i and n i is found to vanish, in general. To understand the importance of rotational inertia and softness of the disks on the spatio-temporal properties, in Fig. 7a, we show the velocity auto-correlation 42 �v i (t) ·v i (t = 0)� of the disks for several values of rotational inertia J, shown in the legend, for fixed κ = 7 , M = 0.05 , and P e = 125.
v i implies v i /|v i | , and ... denotes the average over the system size. In the inset, we show the auto-correlation of the self-propulsion direction �n i (t) · n i (t = 0)� for the same values of J and other parameters, as in Fig. 7a. With increase in J, time correlation of the self-propulsion direction n i increases. Hence, J increases the effective persistence time τ e P of the disks. During head-on collision of two soft self-propelled disks, both the disks deform their shapes substantially and remain attached together on average for their persistence time scale τ P = 1/D r . On the other hand, if they collide with finite impact parameter, that is with an inter-particle distance less than the diameter of the disks, there is a finite probability for the disks to deform their shapes and move apart without much changing their self-propulsion direction 21 . Hence, the time required to remain attached decreases significantly for the soft disks, and their probability to collide with the other disks increases. Consequently, the number of collisions and the collision frequency increases with increase in J for the soft disks, as J increases the effective We define the spatial correlation of velocity 37 of the disks by the relation C(r) = Here, r is the magnitude of the inter-particle distance r ij between the disks at r i and at r j . In Fig. 8a, we show C(r), time averaged over 50 steps in the interval of δt = 1 , in the phase separated state for several values of J in the range [1,10], as shown by the direction of the arrow, for the soft disks at κ = 7 , M = 0.05 , and P e = 125.
In the inset, we show C(r) for increased hardness ( κ = 14 ) over the full range of rotational inertia J = [0.0001, 10] , as the system phase separate at J as low as J = 0.0001 . With increase in J, that is with increase in effective persistence time τ e P , spatial correlation of velocity of the self-propelled disks increases both for the soft and substantially hard disks, which is consistent with the previous work in the respective limits 22,37,41,44 Furthermore, the the strength of the correlation or the correlation length increases with disk hardness κ . However, spatial correlation of the self-propulsion direction n i has not been observed for our system. In Supplementary  Fig. S6, we show the configuration of the system, colored according to the angle of velocity θ(v i ) (top row) and self-propulsion direction θ(n i ) (bottom row) with respect to the x-axis, for three different values of J in the phase separated state. To quantify spatial correlation of velocity or the spatial correlation length, we define a quantity S, which we call strength of the spatial correlation of velocity. To calculate S, we fit C(r) with the expression 41 : m e −ar r b + c and obtain the area under the fit curve using trapezoidal rule of numerical integration in the limit r = 0.1 to r = 50 , that is S = 50 0 C(r + �r)�r . Fix expression for C(r), which is inspired from a similar nature of spatial velocity correlation of hard disks in Ref. 41, is generalised in our work to incorporate the effect of J and κ . The generalized expression fits extremely well with our numerical data. In Fig. 8b, we show S as a function of J for fixed κ = 7 , M = 0.05 , and P e = 125 . In the inset, we show S as a function of disk softness κ , for fixed rotational inertia J = 2 . The other parameter values are same as in Fig. 8b. Hence, strength of the spatial correlation of velocity or the spatial correlation length increases with increase in rotational inertia, while it decreases with increase in disk softness (decrease in κ ). In Supplementary Fig. S7, we show the configuration, colored according to the angle of the velocity θ(v i ) with respect to the x-axis, for three different values of κ at fixed J.

Conclusion
We investigate the combined effect of rotational inertia and disk softness on the collective behavior of soft selfpropelled inertial disks. Several important aspects have been addressed. We report the MIPS phase diagram in the space of rotational inertia of the disks and particle softness over a range of translational inertia and strength of self-propulsion, measured by the Peclet number. The phase boundary between homogeneous (no-MIPS) and MIPS state is found to shift towards larger disk softness with increase in rotational inertia. It was observed that in the limit of negligibly small rotational inertia, increase in softness disfavors MIPS 21 . We demonstrate that rotational inertial dynamics favors MIPS and opens up a new phase space for MIPS, as the system of much softer disks is found to exhibit MIPS in the limit of large rotational inertia. Shift in the phase boundary towards larger disk softness is shown to be the consequence of increased effective persistence of the self-propelled disks with increase in rotational inertia. Our analytical scaling Eq. (6) of the phase boundary are robust, as they are shown to agree with the numerical data over a large range of translational inertia of the self-propelled disks. Although we demonstrate the MIPS phase diagram in the space of rotational inertia and disk softness over a large range of translational inertia and self-propulsion strength, the nature of the non-equilibrium phase transition is an interesting open question of importance that can be attempted in future.
We demonstrate distinct structural and dynamical properties near and away from the phase boundary over a large range of translational inertia. The shape of the high density phase is found to be circular near the phase www.nature.com/scientificreports/ boundary, whereas it is found to be rectangular far away from the phase boundary, which is obtained either by increasing particle softness or rotational inertia. We quantify structural properties by measuring local area fraction and local orientational order of the disks, while dynamical properties are quantified by measuring local speed of the particles. Both structural and dynamical properties are found to remain constant in the central region of the high density phase, whereas they vary gradually near the peripheral region. Near the phase boundary, the peripheral region is shown to be much wider compared to narrow peripheral region away from the phase boundary. Also, the disks are found to infiltrate towards the center of the high density phase, as the local speed of the particles are finite in the high density phase, whereas away from the phase boundary, the particles are not found to move much inside the high density phase, as the local speed are nearly zero, as in conventional overdamped limit of hard self-propelled particles. The disk dynamics are quantified by measuring the fraction of particles retained in the central and the peripheral region of the high density phase. Spatial correlation of velocity of the disks in the high density phase is found to increase with both rotational inertia and particle hardness, whereas temporal correlation of the velocity of the disks in the high density phase is found to only vary with rotational inertia. Temporal correlation of the velocity of the disks in the high density phase is independent of disk softness.
In this work, we consider a rectangular simulation box of fixed dimensions with periodic boundaries. It is likely that the shape and the boundaries of the simulation box would have a nontrivial impact on the properties of the clusters of the soft self-propelled disks, and this may be an interesting area for future studies.
We conjecture that our results should be observed in experiments with soft inertial balls with an appropriate self-propulsion mechanism 5 , in analogy with experiments of vibrated granular particles 25,45 . It would be crucial to perform such experiments in future work. We believe that our investigations are extremely important to extend the understanding of the collective properties of soft self-propelled particles and help in designing improved artificial self-propelled particles that could find its potential in several bio-medical applications 46,47 . Investigation of self-propelled particles with time dependent rotational inertia, as observed in several biological 36 and synthetic 5 active matter, would be interesting to perform.